Heisenberg inequalities for Jacobi transforms (Q881999)

The classical Heisenberg uncertainty principle states that for \(f \in L^2({\mathbb R})\) \[ \biggl(\int_{{\mathbb R}}x^2|f(x)|^2 \,dx \biggr)\cdot \biggl(\int_{{\mathbb R}}\xi^2|\widehat{f}(\xi)|^2\,d\xi\biggr) \geq \frac{1}{4}\|f\|^4_2 \] where \(\widehat{f}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{{\mathbb R}}f(x)e^{-i\xi x}\,dx\). In this paper, the author obtains this inequality and variants of it for the Jacobi transform. The proof is based on ultracontractive properties of the semigroups generated by the Jacobi differential operator and on the estimate of the heat kernel. Let \(\lambda \in {\mathbb C}, \;\alpha \geq \beta \geq -\frac{1}{2},\;\alpha \neq -\frac{1}{2}\). The Jacobi functions \(\phi_{\lambda}\) of order (\(\alpha\), \(\beta\)) is the unique even \(C^{\infty}\)-solution of the differential equation \[ (\mathcal{L}_{\alpha,\beta}+ \lambda^2 + \rho^2)v=0, \;\;v(0)=1, \] where \(\rho =\alpha +\beta +1\) and \(\mathcal{L}_{\alpha, \beta}\) is the Jacobi differential operator \[ \mathcal{L}_{\alpha, \beta}=\frac{d^2}{dx^2}+ \biggl(\frac{\Delta'_{\alpha, \beta}(x)}{\Delta_{\alpha,\beta}(x)} \biggr)\,\frac{d}{dx} \] with \(\Delta_{\alpha,\beta}(x)=(2\sinh x)^{2\alpha +1}(2 \cosh x)^{2\beta +1}\). Let \(L^p({\mathbb R}^{+}, \Delta_{\alpha, \beta}(x)\,dx)\), \(1 \leq p < \infty\), be the space of even functions \(f\) on \({\mathbb R}\) such that \[ \|f\|_{p, {\mathbb R}^{+}}= \left(\int_{0}^{\infty}|f(x)|^p \Delta_{\alpha, \beta}(x) \,dx \right)^{1/p} < \infty. \] Let \(\|f\|_{p,c}= (\frac{1}{2\pi}\int_{0}^{\infty}|f(\lambda)|^p |c(\lambda)|^{-2} \,d\lambda)^{1/p}\) with \[ c(\lambda)= \frac{2^{\rho-i\lambda}\Gamma(\alpha+1) \Gamma(i\lambda)}{\Gamma(\frac{1}{2}(i\lambda +\rho)) \Gamma(\frac{1}{2}(i\lambda+\alpha-\beta +1))}. \] We define the Jacobi transform \(f \rightarrow \widehat{f}^{\alpha, \beta}\) by \[ \widehat{f}^{\alpha, \beta}(\lambda)=\int_{0}^{\infty}f(x)\phi_{\lambda}(x) \Delta_{\alpha, \beta}(x) \,dx \] for all functions \(f\) on \({\mathbb R}^+\) and complex numbers \(\lambda\) for which the right-hand side is well defined. Let \(D_{\alpha}=[\frac{1}{2}, 1]\) if \(\alpha \geq \frac{1}{2}\). For \(-\frac{1}{4} < \alpha <\frac{1}{2}\), if \(\rho \geq 1\), \(D_{\alpha}=[\frac{1}{2}, \frac{2(\alpha+1)}{3}]\), otherwise \(D_{\alpha}=(\frac{1}{2}, \frac{2(\alpha+1)}{3}]\). Theorem A. (Uncertainty principle for the Jacobi transform). For \(\alpha \geq \beta \geq -\frac{1}{2}\), \(\alpha > -\frac{1}{4}\), assume \(a,b >0\) and \(\gamma \in D_{\alpha}\). Then there exists a constant \(C>0\) such that \[ \|x^{\gamma a}f\|^{\frac{b}{a+b}}_{2, {\mathbb R}^+}\|(\lambda^2+\rho^2)^{b/2} \widehat{f}^{\alpha,\beta}\|_{2,c}^{\frac{a}{a+b}} \geq C\|f\|_{2,{\mathbb R}^+} \] for all \(f \in L^2({\mathbb R}^+ , \Delta_{\alpha, \beta}(x)dx)\). The functions \(\psi_{\lambda}\) are the unique \(C^{\infty}\)-solution of the differential-difference equation \[ (\mathcal{I}_{\alpha, \beta}+i\lambda)u=0,\;\;u(0)=1 \] where \[ \mathcal{I}_{\alpha, \beta}f(x)=\frac{d}{dx}f(x)+ \frac{\Delta'_{\alpha,\beta}(x)}{\Delta_{\alpha, \beta}(x)} \frac{f(x)-f(-x)}{2}. \] Let \(L^p({\mathbb R}, \Delta_{\alpha,\beta}(|x|)dx), \;1 \leq p < \infty \;\), be the space of functions \(f\) on \({\mathbb R}\) such that \[ \|f\|_{p, {\mathbb R}}= \Biggl(\int_{{\mathbb R}}|f(x)|^p \Delta_{\alpha, \beta}(|x|)\,dx \Biggr)^{1/p}< \infty. \] Let \(||f||_{p, \sigma}=(\int_{{\mathbb R}}|f(\lambda)|^p d\sigma(\lambda))^{1/p}\) and \(d\sigma\) is the measure given by \[ d\sigma(\lambda)= \frac{|\lambda|} {8\pi \sqrt{\lambda^2-\rho^2}|c(\sqrt{\lambda^2-\rho^2})|^2} \chi_{{\mathbb R}\setminus (-\rho, \rho)}(\lambda)d\lambda\, \] where \(\chi_{{\mathbb R}\setminus (-\rho, \rho)}\) is the characteristic function of \({\mathbb R} \setminus (-\rho, \rho)\). We define the Jacobi-Dunkl transform \(f \rightarrow \mathcal{F}f\) by \[ \mathcal{F}f(\lambda)= \int_{\mathbb R}f(x)\psi_{\lambda}(x) \Delta_{\alpha, \beta}(|x|)\,dx. \] Theorem B. (Uncertainty principle for the Jacobi-Dunkl transform). For \(\alpha \geq \beta \geq -\frac{1}{2},\;\alpha > -\frac{1}{4}\), assume \(a,b >0\) and \(\gamma \in D_{\alpha}\). Then there exists a constant \(C>0\) such that \[ |||x|^{\gamma a}f||^{\frac{b}{a+b}}_{2, {\mathbb R}} \| |\lambda|^b \mathcal{F}f\|_{2,\sigma}^{\frac{a}{a+b}} \geq C\|f\|_{2, {\mathbb R}} \] for all \(f\in L^2({\mathbb R}, \Delta_{\alpha,\beta}(|x|)\,dx)\).